WASHINGTON -- Republicans left out many details from the tax-code outline they released last week, and few are more important for middle-income households than the fate of the child tax credit.

They plan to bump up the $1,000 per-child credit for children under age 17, but they haven't settled on how much or at what income levels the credit would start to shut off. There is a divide in the party over the issue. Family advocates want to boost the child tax credit as much as possible; others would rather focus tax cuts on promoting faster economic growth.

Where they settle will determine whether millions of middle-income households get a net tax cut or whether some end up paying more. It also will help shape the broader tax policy debate in the weeks and months ahead.

"The tax framework, the way it's laid out, makes the entire thing hinge on the child tax credit," said Samuel Hammond, an advocate for the credit who is an analyst at the libertarian Niskanen Center.

The debate over the credit also is a window into the trade-offs Republicans are weighing as they try to make good on promises of a booming economy and middle-class tax cuts.

"My preference is to focus on pro-growth policies," Sen. Pat Toomey (R., Pa.) said. The child tax credit will be expanded, he said, but it "does not encourage economic growth." Lower tax rates, especially for businesses, would be more effective, he and others in the GOP's growth-first wing argue.

Senators Mike Lee (R., Utah) and Marco Rubio (R., Fla), with support from Ivanka Trump, are pushing a $2,000 credit per child, double the current amount. Their formula for calculating the credit is more generous to low-income families than current law is, and it would be refundable, meaning people can get money back from the Internal Revenue Service even if they don't owe income taxes.

The GOP tax plan for now doesn't have enough detail to show how many middle-income households with children would benefit.

A Tax Policy Center analysis released last week found that the GOP's framework would increase taxes on 17% of households with children in 2018 and 44% by 2027. Those estimates hang on many assumptions, including about the size and nature of the expanded child credit and the income cutoffs for new tax brackets. Republicans dispute the analysis because those levels haven't been set; the nonpartisan center is run by a former Obama administration official.

Still, the figures in the analysis raised alarm bells on the left -- and on the right.

"This is a GOP tax plan?" Sen. Rand Paul (R., Ky.) said on Twitter this week. "Possibly 30% of middle class gets a tax hike? I hope the final details are better than this."

The Tax Policy Center study assumed a $1,500 credit -- the current $1,000 credit for children under age 17 plus a new $500 nonrefundable credit per child like one House Republicans proposed last year. By making it nonrefundable, Republicans wouldn't make that portion of the credit available to households that don't pay income taxes. A similar credit would be available for dependents who aren't children.

An expanded child tax credit would help offset another change Republicans plan: repealing the personal exemption in the name of simplifying the tax code. Today, the personal exemption subtracts $4,050 from income for each person in the household, which would cut the tax bill of someone in the new bottom bracket of 12% by as much as $486 per child. That is roughly equivalent to a $500 increase in the credit.

The increase in the child credit will be larger than the $500 credit that the Tax Policy Center assumed and will be designed to do more than just offset the loss of the personal exemption, said an administration official. Without including a number, the tax framework released last week by top negotiators from the House, Senate and White House assumed a child tax credit larger than $1,500 but smaller than the $2,000 that senators Lee and Rubio support. The negotiators didn't agree on whether to make the expanded portion of the credit refundable, the official said.

Republicans also plan to increase the income level at which the credit begins phasing out, now at $75,000 for individuals and $110,000 for married couples filing jointly, the official said. That means more households in the upper middle class could benefit, though the thresholds haven't been set.

Another key question is whether those figures -- the credit amount and the income levels -- will be indexed to inflation. Right now, they aren't, eroding the benefit of the child tax credit over time. The TPC analysis doesn't assume that the expanded part of the credit will be indexed.

The more generous Lee-Rubio proposal for a $2,000 refundable credit heads in the right direction but still falls short of helping families left behind in the modern economy, said Chuck Marr, director of federal tax policy at the left-leaning Center on Budget and Policy Priorities.

"If one is going to move around trillions of dollars, helping those people should be a top priority," he said.

If the expanded credit is $500, it won't offset the loss of personal exemptions for households in the 25% tax bracket, because each personal exemption now is worth more than $1,000 per person in tax savings. That and other features could lead to tax increases for middle-income families, though many households would also benefit from a larger standard deduction.

The exact details will be set by the House Ways and Means and Senate Finance committees, which don't plan to release them for weeks.

Especially in the Senate, they will face a vote-counting challenge. Republicans have 52 seats and any three senators -- say, Messrs. Lee and Rubio and one other -- have the clout to insist on changes.

Sen. Rob Portman (R., Ohio) said Tuesday that he supports the direction Messrs. Lee and Rubio are headed in.

The child tax credit is a centerpiece of the so-called "reformicon" movement, the strand of GOP thinking that puts more emphasis on direct benefits to families and less on cutting marginal tax rates. Compared with reductions in tax rates or deductions, credits are more valuable to lower-income and middle-income households.

Unlike the version of the break President Donald Trump talked about during his campaign, the child credit envisioned in the framework isn't tied to child care costs, making it available to stay-at-home parents and working parents.

"It delivers resources to families, including lower-income families, in a way that doesn't dictate lifestyle choices," Mr. Hammond said.

The populist notes were already evident in the GOP rhetoric about their tax plan. Mr. Trump has said repeatedly that he wants to focus on a middle-class tax cut.

But trade-offs abound. Doubling the child tax credit as Messrs. Lee and Rubio want would cost the government about $500 billion over a decade in foregone revenue. That is about double the cost of the increase that the Tax Policy Center assumed. Making room for a larger child tax credit would require other tax breaks to go away or reductions in tax rates to get scaled back.

"It's a really high priority," said Sen. John Thune (R., S.D.), a Finance Committee member who said he'd like the credit as close to the Lee-Rubio numbers as budget constraints allow. "It will be as big and robust as we can possibly can possibly make it."